[SciPy-User] [OT] Transform (i.e., Fourier, Laplace, etc.) methods in Prob. & Stats.

[josef.pktd at gmail.com]

On Wed, Nov 25, 2009 at 7:31 PM, David Goldsmith
<d.l.goldsmith at gmail.com> wrote:
> On Wed, Nov 25, 2009 at 3:45 PM, <josef.pktd at gmail.com> wrote:
>>
>> On Wed, Nov 25, 2009 at 6:24 PM, David Goldsmith
>> <d.l.goldsmith at gmail.com> wrote:
>> > Good info, thanks; I'll look up "your" thread, Josef, on the archive and
>> > run
>> > down what look like relevant references.  (FWIW, my interest is that I'm
>> > helping out (nominally, "tutoring," but this level, it's more akin to
>> > being
>> > a sounding board, checking his derivations, and "reminding" him of
>> > various
>> > subtleties that are emphasized in math, but not necessarily in EE, etc.)
>> > this guy working on his dissertation on air traffic control automation
>> > using
>> > wireless communication protocols, very probability heavy stuff, and for
>> > the
>> > second time yesterday, he presented me with a transform application - in
>> > this instance, the "Z" transform - in this probability-heavy stuff, and
>> > this
>> > is outside of my training in probability, so I want to "bone-up.")
>> > Thanks
>> > again,
>>
>> I always have to look for your reply because you don't follow our
>> bottom-posting
>> policy.
>
> Sorry, I tend to "follow" when I'm saying something in direct response to
> something I'm replying to and/or when I think that I'm likely _not_
> terminating the thread, but when I'm responding generally and/or think that
> I am likely terminating the thread, then I tend to just reply at the top.
> I'll try to remember that we have a policy. :-)
>
>> I have seen the z-transform only in the context of time series analysis
>> http://en.wikipedia.org/wiki/Z-transform
>> especially this
>>
>> http://en.wikipedia.org/wiki/Z-transform#Linear_constant-coefficient_difference_equation
>> covered to some extend in scipy.signal, lfilter and lti
>
> Part of the problem was that it wasn't clear to either of us - myself or my
> "student" - why the authors of this particular paper were using the
> z-transform at all where they were - it seemed their result was easily
> derivable w/out it, so we were both baffled.
>
>> so the other literature to Laplace transforms and characteristic functions
>> might not be very closely related.
>
> Perhaps not directly (in any event, presently, I'm interested in
> theoretical/"analytical," i.e., not numerical, applications anyway), but my
> philosophy has always been, if I can be directed to something that is closer
> to on target than what I've been able to find on my own, then even if it's
> not a bulls-eye, I can often find a bulls-eye in the reference's
> references.  For example, "Chung (or any other book on graduate
> probability)" sounds like a good starting point.  So thanks for reminding me
> about the thread.  (I knew it sounded familiar: I contributed to it!  And on
> that note, I "let it lie" at the time, but now feel I should say, admittedly
> a little defensively, that of course Anne's comments were on the mark; the
> only reasons I felt it necessary to add what I did about complex integration
> over a closed path were: A) you had indicated that you were a bit of a
> novice in the field, and the result I was giving is, perhaps arguably, the
> subject's most fundamental result, and B) I felt that it was important that
> you were aware of it because, if any of your functions _were_ analytic and
> your paths closed, then you shouldn't be doing any (explicit) numerical (or
> symbolic, for that matter) integration at all - you should just be
> "hard-wiring" those integrals to zero!  And for what it's worth: every time
> you integrate with respect to one (continuous) real variable, you're doing a
> path integration - one so comparatively trivial that we don't call it that,
> but a path integration nevertheless.) :-)

I was just reading up a bit on contour integrals on wikipedia, and it
looks too applied for Probability and Measure theory. It just tells
you how to use some tricks to calculate specific Rieman integrals in
the complex plane. I didn't see any hints for Lebesque integrals.
All real analysis, and measure theory (that I have seen) is based on
Lebesque integration or Lebesque-Stiltjes as in Chungs book. So for me
contour integrals just falls in between the measure theory and the
applied (real) calculations, and I never had to figure out what it
does.

I'm not doing path integration when I integrate with respect to a
(probability) measure that has both continuous intervals and mass
points (Lebesque not Rieman if you want to be picky)

Josef




>
> DG
>
>>
>> Josef
>>
>>
>> >
>> > DG
>> >
>> > On Wed, Nov 25, 2009 at 2:41 PM, nicky van foreest
>> > <vanforeest at gmail.com>
>> > wrote:
>> >>
>> >> Hi,
>> >>
>> >> 2009/11/25  <josef.pktd at gmail.com>:
>> >> > On Wed, Nov 25, 2009 at 2:53 PM, David Goldsmith
>> >> > <d.l.goldsmith at gmail.com> wrote:
>> >> >> Are there enoug applications of transform methods (by which I mean,
>> >> >> Fourier, Laplace, Z, etc.) in probability & statistics for this to
>> >> >> be
>> >> >> considered its own specialty therein?  Any text recommendations on
>> >> >> it
>> >> >> (even
>> >> >> if it's only a chapter dedicated to it)?  Thanks,
>> >> >>
>> >> >
>> >> > Some information is in the thread on my recent question
>> >> > "characteristic functions of probability distributions"
>> >> >
>> >> > There is a large literature in econometrics and statistics about
>> >> > using
>> >> > the characteristic function for estimation and testing.
>> >> > The reference of Nicky for queuing theory uses mostly the Laplace
>> >> > transform (for discrete distributions),
>> >>
>> >> It has been some time ago (more than 5 years...), but I recall that
>> >> Whitt, in his articles on the numerical inversion of Laplace
>> >> transforms, discretized Laplace transforms to facilitate the
>> >> inversion, The distributions themselves are not necessarily discrete.
>> >> One example would be the waiting time distribution of customers in a
>> >> queue, which is continuous for most service and arrival processes.
>> >>
>> >> There is certainly potential for dedicated numerical inversion algo's
>> >> for the Laplace transforms of density and distribution functions. The
>> >> latter form a somewhat specialized sort of function. Distribution
>> >> functions are 0 at -\infty, and 1 at \infty, and are non decreasing.
>> >> They may also have discontinuities, but not too many. These properties
>> >> may affect the inversion.  Besides these properties, the transforms
>> >> are used to obtain insight into the behavior of the sum of independent
>> >> random variables. Such sums can be rewritten as the product of the
>> >> transforms of distribution. This product in turn requires inversion
>> >> to, as some people call it, take away the Laplacian curtain.
>> >>
>> >> Nicky
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